Transformations of states in quantum mechanics

Question

In Classical Mechanics we usually describe the possible configurations of a system by points on a smooth manifold $M$ which is the configuration manifold of the system.

In that case, when we talk about transformations of the possible configurations of the system we usually do so introducing diffeomorphisms and one-parameter families of diffeomorphisms. The infinitesimal version of those transformations are then described by vector fields on the configuration manifold. The generator of a one-parameter family of transformations is then a vector field.

Now, in Quantum Mechanics we describe the possible states of the system by vectors in a Hilbert space $\mathcal{H}$. I'm unsure, however, how do we describe those "transformations" (for example, translations and rotations) like we do in Classical Mechanics.

Since we always want to preserve the linear structure it is clear those transformations should be linear operators in $\mathcal{H}$. But it is only this that we require?

Also, how does one describe the "generator of a certain transformation" as we do in Classical Mechanics? We don't have vector fields here to do this, so I'm unsure how to describe it.

In that case, from a mathematically rigorous standpoint, how does one, in analogy with Classical Mechanics, describe transformations of states corresponding to actual transformations of the systems? And how do we describe the infinitesimal version of those transformations properly?

Answer 1

Quantum mechanical transformations have to be implemented as unitary operators since a transformation should not change the norm of a state, for instance. In particular, Wigner's theorem shows that every symmetry transformation must be given by a unitary or anti-unitary operator.

Thus, the proper way to talk about families of transformations in quantum mechanics is Stone's theorem on one-parameter unitary groups stating that, to every self-adjoint operator $T$ on a Hilbert space $H$ there is a strongly continuous family of unitary transformations $U_T(t):= \mathrm{e}^{\mathrm{i}Tt}\in\mathrm{U}(H),t\in\mathbb{R}$ and that, conversely, every such strongly continuous family has a corresponding self-adjoint generator.

The most familiar instance of such a correspondence is the time evolution $U(t)$ which is generated as $\mathrm{e}^{\mathrm{i}Ht}$ by the Hamiltonian $H$, which signifies an infintesimal time evolution exactly as in classical Hamiltonian mechanics.

The classical generators of transformations become observables on the quantum space of states in the process of quantization (or are defined as observables, if we are not quantizing), and they there generate their corresponding transformations by Stone's theorem.